To solve the equation \(2x\sqrt{2x+3} = 5x^2 + 6x + 1\), we can start by isolating the square root term and then squaring both sides to eliminate the square root.

1. Rearranging the equation:
\[
2x\sqrt{2x+3} = 5x^2 + 6x + 1
\]

2. Divide both sides by \(2x\) (assuming \(x \neq 0\)):
\[
\sqrt{2x+3} = \frac{5x^2 + 6x + 1}{2x}
\]

3. Now, square both sides:
\[
2x + 3 = \left(\frac{5x^2 + 6x + 1}{2x}\right)^2
\]

4. To simplify, expand the right-hand side:
\[
\left(\frac{5x^2 + 6x + 1}{2x}\right)^2 = \frac{(5x^2 + 6x + 1)^2}{4x^2}
\]

5. So the equation becomes:
\[
2x + 3 = \frac{(5x^2 + 6x + 1)^2}{4x^2}
\]

6. After this, you would typically multiply both sides by \(4x^2\) to eliminate the fraction and simplify further.

Continue solving from this point by expanding the squared term and rearranging to form a polynomial equation, then solve for \(x\) using factoring, the quadratic formula, or another method as necessary. Make sure to check for extraneous solutions after finding the values of \(x\).